On the asymptotic behavior of solutions of linear difference equations (Q1344456)

For linear difference equations with advanced arguments of the form \[ \Delta x_n= \sum^r_{i=0} a_n^{(i)} x_{n+i}, \quad n\in \mathbb{N} \] sufficient conditions are presented which guarantee that for any real constant \(C\neq 0\) there exists a solution of this equation such that \(x_n= C+ o(1)\) as \(n\to \infty\). This result is applied to \(m\)-th order equations of the form \(\Delta^m x_n= a_n x_n\), \(n\in \mathbb{N}\) where for any \(C\neq 0\) this solution has the asymptotic behavior \[ x_n= \Biggl[m^{-n} \prod^{n-1}_{j=1} [1+ (-1)^{m+ 1} a_j]\Biggr](C+ o(1)) \quad \text{ as } n\to \infty. \]